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This thesis consists of three chapters on the statistical adjustment, calibration, and uncertainty quantification of complex computer models with applications in engineering. The first chapter systematically develops an engineering-driven statistical adjustment and calibration framework, the second chapter deals with the calibration of potassium current model in a cardiac cell, and the third chapter develops an emulator-based approach for propagating input parameter uncertainty in a solid end milling process.
Engineering model development involves several simplifying assumptions for the purpose of mathematical tractability which are often not realistic in practice. This leads to discrepancies in the model predictions. A commonly used statistical approach to overcome this problem is to build a statistical model for the discrepancies between the engineering model and observed data. In contrast, an engineering approach would be to find the causes of discrepancy and fix the engineering model using first principles. However, the engineering approach is time consuming, whereas the statistical approach is fast. The drawback of the statistical approach is that it treats the engineering model as a black box and therefore, the statistically adjusted models lack physical interpretability. In the first chapter, we propose a new framework for model calibration and statistical adjustment. It tries to open up the black box using simple main effects analysis and graphical plots and introduces statistical models inside the engineering model. This approach leads to simpler adjustment models that are physically more interpretable. The approach is illustrated using a model for predicting the cutting forces in a laser-assisted mechanical micromachining process and a model for predicting the temperature of outlet air in a fluidized-bed process.
The second chapter studies the calibration of a computer model of potassium currents in a cardiac cell. The computer model is expensive to evaluate and contains twenty-four unknown parameters, which makes the calibration challenging for the traditional methods using kriging. Another difficulty with this problem is the presence of large cell-to-cell variation, which is modeled through random effects. We propose physics-driven strategies for the approximation of the computer model and an efficient method for the identification and estimation of parameters in this high-dimensional nonlinear mixed-effects statistical model.
Traditional sampling-based approaches to uncertainty quantification can be slow if the computer model is computationally expensive. In such cases, an easy-to-evaluate emulator can be used to replace the computer model to improve the computational efficiency. However, the traditional technique using kriging is found to perform poorly for the solid end milling process. In chapter three, we develop a new emulator, in which a base function is used to capture the general trend of the output. We propose optimal experimental design strategies for fitting the emulator. We call our proposed emulator local base emulator. Using the solid end milling example, we show that the local base emulator is an efficient and accurate technique for uncertainty quantification and has advantages over the other traditional tools.